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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJan 4th 2011
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   Author: zskoda
   Format: MarkdownItexI have created a stub for [[constructible universe]]. I did not go through the version of the definition via definability. Now constructible sets are sets in the constructible universe. The notion of course, intentionally reminds the constructible sets in topology and algebaric geometry as exposed e.g. in the books on stratified spaces, on perverse sheaves (MacPherson e.g.) and in Lurie's [[Higher Topos Theory]]. Now I wanted to create [[constructible set]] but I was hoping that there is a common definition for all these cases or at least logically defendable unique point of view, rather than partial similarity of definitions. I mean one always have some business of unions, complements etc. starting with some primitive family, say with open sets, or algebraic sets, or open sets relative strata etc. and inductively constructs more. Now, all the operations mentioned seem to have sense in some class of lattices. Maybe in [[Heyting lattice]]s or at least in Boolean lattices. On the other hand, google spits out several references on *constructible lattices* *one of the authors is certain Janowitz), but the definition there is disappointing. I mean I would like that one has some sort of constructible completion of certain kind of a lattice and talk about the constructible elements as the elements of constructible completion. I am sure that the nLab community could nail the wanted common generalization down or to give a reference if the literature has it already.

   I have created a stub for constructible universe. I did not go through the version of the definition via definability. Now constructible sets are sets in the constructible universe. The notion of course, intentionally reminds the constructible sets in topology and algebaric geometry as exposed e.g. in the books on stratified spaces, on perverse sheaves (MacPherson e.g.) and in Lurie’s Higher Topos Theory. Now I wanted to create constructible set but I was hoping that there is a common definition for all these cases or at least logically defendable unique point of view, rather than partial similarity of definitions. I mean one always have some business of unions, complements etc. starting with some primitive family, say with open sets, or algebraic sets, or open sets relative strata etc. and inductively constructs more. Now, all the operations mentioned seem to have sense in some class of lattices. Maybe in Heyting lattices or at least in Boolean lattices. On the other hand, google spits out several references on constructible lattices *one of the authors is certain Janowitz), but the definition there is disappointing. I mean I would like that one has some sort of constructible completion of certain kind of a lattice and talk about the constructible elements as the elements of constructible completion. I am sure that the nLab community could nail the wanted common generalization down or to give a reference if the literature has it already.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 6th 2011
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   Author: zskoda
   Format: MarkdownItexNew stub [[Morse-Kelley set theory]].

   New stub Morse-Kelley set theory.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 7th 2011
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   Author: DavidRoberts
   Format: MarkdownItexThere is a little bit about MK already at [[ZFC]], Zoran, but it could do with its own page, I agree. I added a few links to the stub at [[Morse-Kelley set theory]].

   There is a little bit about MK already at ZFC, Zoran, but it could do with its own page, I agree.

   I added a few links to the stub at Morse-Kelley set theory.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJan 7th 2011
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   Author: zskoda
   Format: MarkdownItexTo David/3 Well, the problem was that only the abbreviation was there (I often protest that we should not use slang too much, including abbreviations) without *ever* the full name. Thus the search for Morse-Kelley in $n$Lab at the time of me creating a new entry did not show *any* hits. Thank you for linking now and mentioning that there is a statement there. There should be more about it...

   To David/3

   Well, the problem was that only the abbreviation was there (I often protest that we should not use slang too much, including abbreviations) without ever the full name. Thus the search for Morse-Kelley in $n$Lab at the time of me creating a new entry did not show any hits. Thank you for linking now and mentioning that there is a statement there. There should be more about it…

5. • CommentRowNumber5.
   • CommentAuthorGuest
   • CommentTimeAug 23rd 2011
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   Author: Guest
   Format: TextI think that your problem with constructible sets is that there is no agreed meaning for the term. There are relative terms like 'constructible from' rather than an absolute term 'constructible'. Your function [P(x) intersection I(X')] gives the 'sets directly constructible from X. Your construction L, which iterates the construction transfinitely often give the sets hereditarily constructible from X. Godels's universe is L(0). Solovay et al have considered universes such as L(R) [=the universe of sets hereditarily constructible from the reals]. In a sense, these universes are the smallest universes containing the specified base set. Also your page on MK should probably include the fact that MK is equivalent to NBG + 'There exists an inaccessible ordinal. Roger Witte

   I think that your problem with constructible sets is that there is no agreed meaning for the term. There are relative terms like 'constructible from' rather than an absolute term 'constructible'. Your function [P(x) intersection I(X')] gives the 'sets directly constructible from X. Your construction L, which iterates the construction transfinitely often give the sets hereditarily constructible from X. Godels's universe is L(0). Solovay et al have considered universes such as L(R) [=the universe of sets hereditarily constructible from the reals]. In a sense, these universes are the smallest universes containing the specified base set.
   
   Also your page on MK should probably include the fact that MK is equivalent to NBG + 'There exists an inaccessible ordinal.
   
   Roger Witte
6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 23rd 2011
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   Author: Mike Shulman
   Format: MarkdownItex> Also your page on MK should probably include the fact that MK is equivalent to NBG + 'There exists an inaccessible ordinal. I presume that by "equivalent to" you mean "equiconsistent with"? I think "NBG + there exists an inaccessible (set) ordinal" would still not let you do the things with proper classes that MK does; but I believe I've read that if $\kappa$ is inaccessible, then there is a model of MK with $V_\kappa$ = sets, $V_{\kappa+1}$ = classes. But it's surprising to me to hear that MK is as strong as the existence of an inaccessible; can you give a reference?

   Also your page on MK should probably include the fact that MK is equivalent to NBG + ’There exists an inaccessible ordinal.

   I presume that by “equivalent to” you mean “equiconsistent with”? I think “NBG + there exists an inaccessible (set) ordinal” would still not let you do the things with proper classes that MK does; but I believe I’ve read that if $\kappa$ is inaccessible, then there is a model of MK with $V_\kappa$ = sets, $V_{\kappa+1}$ = classes. But it’s surprising to me to hear that MK is as strong as the existence of an inaccessible; can you give a reference?